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Just to clarify, by harmonic you mean: $$(d\delta+\delta d)F=0$$ where d is the exterior derivative and delta is the co-dfifferential?
Is this true? It seems since dF=0 by definition, then we need to show: $$d\delta F=-d(*d*F)=4\pi(d**J)=0$$
Is it true that $$dJ=0$$? I can't think of a reason for this...